Andrew DeLapo (UConn) – Index Sets and Computable Categoricity of CSC Spaces

Given a topology on the natural numbers, how complicated is it to describe? To answer this question with tools from computability theory, we will restrict to the context of countable second-countable (CSC) topological spaces. One approach is to assign an index to each computable CSC space and determine the arithmetic complexity of the set of CSC spaces with some property. Another approach comes from computable structure theory; for example, given two computable copies of a CSC space, does there exist a computable homeomorphism between them? In this talk, we will explore these approaches and apply them in three running examples: the indiscrete, discrete, and initial segment topologies.